How are the 6 basic trigonometric functions related to right triangles?

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How are the 6 basic trigonometric functions related to right triangles?

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Answer 1 · 2014-11-26T05:49:47

The six basic trigonometric functions are:
1. Sine, $sintheta$
2. Cosine, $costheta$
3. Tangent, $tantheta$
4. Cotangent, $cottheta$
5. Secant, $sectheta$
6. Cosecant, $csctheta$

Take the following triangle for example:

enter image source here

Let the angle marked at A be $theta$ .
The longest side of the triangle is the hypotenuse, the side next to the angle is the adjacent and the side opposite to it is the opposite.

Then,

$sintheta =(opp)/( hyp)=a/h$

$costheta =(adj)/( hyp)=b/h$

$tantheta =(opp)/(adj)=a/b=(sintheta)/(costheta)$

$cottheta =(adj)/(opp)=b/a=(costheta)/(sintheta)=1/tantheta$

$sectheta =(hyp)/(adj)=h/b=1/costheta$

$csctheta =(hyp)/(opp)=h/a=1/sintheta$

Now if we the triangle were to have sides 3, 4 and 5 units long such that
$a=4, b=3, h=5$
The six basic trigonometric functions would be:

$sintheta =a/h=4/5$

$costheta=b/h=3/5$

$tantheta =a/b=4/3$

$cottheta =b/a=3/4$

$sectheta =h/b=5/3$

$csctheta =h/a=5/4$

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How are the 6 basic trigonometric functions related to right triangles?

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